In a model with a scalar singlet, the SM-like Higgs scalar has weaker interactions with fermion and vector bosons. Such interactions are suppressed by a factor of \(c_h\)  relative to SM interactions. The vertices \(hZZ\) , \(hZZ^\prime \)  and \(hZ^\prime Z^\prime \)  have an extra term proportional to \(s_h\) . Only when \(c_h = 1\) , do the interactions with the SM particles for both models agree. However, in the models of Refs. [1]} and [2]},
the dark vector still interacts with the singlet \(h\) . In general, the phenomenology of both models is a bit different in processes like \(h\rightarrow Z^\prime Z,Z^\prime Z^\prime ,ZZ\)   [3]}. This may imply that the region allowed for the four parameters \(m_\eta , g_\eta , g_{VB}\)  and \(m_V\)  is different in both models, although an overlap may also exist, as follows from the DM calculation above. For example, in the models of Refs. [1]} and [2]},
the mass of the vector field \(m_V=g_\eta \textrm {v}_S\)  and the mass of \(h\)  also depends on \(v_S\) . Hence, constraints on \(m_V\)  are also constraint on \(m_h\)  which could be in disagreement with accelerator data. This relation does not exist in our model in which there is no scalar singlet. In our model \(m_V\)  is a free parameter. These interactions are equal to those in the appendix only when \(c_h = 1\)  in equation (REF ).
